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On the long-run fluctuations of inheritance in two-sector

OLG models

Florian Pelgrin, Alain Venditti

To cite this version:

Florian Pelgrin, Alain Venditti. On the long-run fluctuations of inheritance in two-sector OLG models. 2020. �halshs-03080407�

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Working Papers / Documents de travail

WP 2020 - Nr 48

On the long-run fluctuations of inheritance

in two-sector OLG models

Florian Pelgrin Alain Venditti

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On the long-run fluctuations of inheritance in

two-sector OLG models

Florian PELGRIN

EDHEC Business School florian.pelgrin@edhec.edu

and

Alain VENDITTI

†‡

Aix-Marseille Univ., CNRS, AMSE & EDHEC Business School

alain.venditti@univ-amu.fr

This paper is dedicated to Pierre Cartigny (1946-2019) and Carine Nourry (1972-2019)

This work was supported by French National Research Agency Grants ANR-08-BLAN-0245-01 and

ANR-17-EURE-0020. We thank W. Briec, F. Dufourt, J.M. Grandmont, K. Hamada, K. Nishimura, X. Raurich, T. Seegmuller, G. Sorger for useful comments and suggestions, and Elias Chaumeix for his computational assistance during a summer internship at the beginning of this project. This paper also benefited from presentations at the ”18th Journ´ees Louis-Andr´e G´erard-Varet”, Aix-en-Provence, June 13-14, 2019, at the ”19th Annual SAET Conference”, Ischia, June 30-July 6, 2019, at the ”International conference on economics and finance: celebrating Professor Jean-Michel Grandmont’s 80th birthday”, Kobe University, October 14, 2019, and at the Workshop ”Tribute to Carine Nourry”, AMSE, November 8, 2019.

Pierre Cartigny was my PhD advisor. I met him first when I was a student and we became friends

during my PhD. He has been a wonderful teacher and co-author. I will greatly miss him. Carine Nourry was a very close friend. I met her in 1994 as a student during her Master and we started to work jointly during her PhD. I remember as if it was yesterday how we discussed complex dynamics in optimal growth models with heterogeneous agents in our shared office in Marseille. She was a distinguished scholar and a wonderful co-author. I will also greatly miss my dear friend.

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Abstract: This paper provides a long-run cycle perspective to explain the behavior of the annual flow of inheritance as identified by Piketty [51] for France and Atkinson [3] for the UK. Using a two-sector Barro-type [9] OLG model with non-separable preferences and bequests, we show that endogenous fluc-tuations are likely to occur through period-2 cycles or Hopf bifurcations. Two key mechanisms, which can generate independently or together quasi-periodic cycles, can be identified as long as agents are suffi-ciently impatient. The first mechanism relies on the elasticity of intertemporal substitution or equivalently the sign of the cross-derivative of the utility function whereas the second rests on sectoral technologies through the sign of the capital intensity difference across two sectors. Furthermore, building on the quasi-palindromic nature of the degree-4 characteristic equation, we derive some meaningful sufficient conditions associated to the occurrence of complex roots in a two-sector OLG model. Finally, we show that our theoretical results are consistent with some empirical evidence for medium- and long-run swings in the inheritance flows as a fraction of national income in France over the period 1896-2008.

Keywords: Two-sector overlapping generations model, optimal growth, endogenous fluctuations, quasi-palindromic polynomial, periodic and quasi-periodic cycles, altruism, bequest

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1

Introduction

In a very influential contribution, Piketty [51] shows that inherited wealth has again a prominent role for life-cycle income, especially with respect to human capital and labor income. Using a simple generic overlapping generations model of wealth accumulation, growth, and inheritance, Piketty [51] argues that the gap between the (steady-state) growth rate g and the rate of return on private wealth r is the core argument of this resurgence. On the one hand, when g is large and g > r, the new wealth accumulated out of current income, and thus human capital, contributes more to life-cycle income than past inherited wealth, especially when it is grounded on low (past) income relative to today’s income.1 In this respect, inheritance flows remains a small fraction of national income. On the other hand, when growth is low such that r > g, inherited wealth is capitalized at a faster (growth) rate than national income and becomes dominant with respect to current income. Consequently, inheritance flows become a larger proportion of national income. Accordingly, it turns out that the annual bequest flow relative to national income in France would be now close to its steady-state 15%-20%, and will slowly recover its level in the nineteenth century. Furthermore, it provides theoretical foundations on the very pronounced U-shaped pattern of the French inheritance-related variables.2

In fact, the dynamics of the inheritance flow can also be consistent with the predictions of multiple equilibria models and long-run (stochastic) limit cycles.3 Notably the economy can converge to a stable (long-run) cyclical path where some macroeconomic variables oscillate indefinitely around the steady-state, and thus bequest flows can go back and forth to a low, steady or high level. Moreover, when examined over long periods, historical series of inheritance flows (Piketty, [51]) look part of a long cycle through its U-shapped

1For instance, such a pattern is observed in France during the period 1950-1970. 2A similar pattern is found by Atkinson [3] for the UK.

3Deterministic limit cycles have a long tradition in economics. Especially the seminal contributions

of Benhabib and Nishimura [15, 16] have shown that even in standard models featuring forward-looking agents and a competitive equilibrium structure, the steady state or balanced growth path was inherently unstable and thus deterministic (endogenous) fluctuations were easily obtained as soon as the fundamen-tal nonlinear structure of the model was taken into account. More recently, Beaudry et al. [11, 12, 13] have put forward the existence of endogenous stochastic limit cycles, i.e. a deterministic cycle where the stochastic component is essentially an i.i.d. process, that can generate alternate periods of booms and busts (see also Benhabib and Nishimura [17]). Recent strands of the literature that discuss the emergence of limit cycles include contributions on innovation-cycles and growth (Matsuyama [42], Growiec et al. [33]), on endogenous credit cycles in OLG models (Azariadis and Smith [4], Myerson [49], and Gu et al. [34]), on endogenous learning- and bounded rationality-based business fluctuations (Hommes, [35]).

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pattern, especially when extracting its medium to long-run component.4 In a broader

perspective, this descriptive fact is to be reconciled with recent papers (e.g., Beaudry et al. [11, 12, 13]; Growiec et al. [32, 33]) that challenge the seminal contributions of Granger [31] and Sargent [56]: macroeconomic variables do not display (very) pronounced peaks at business cycles frequencies and thus data are not supportive of strong internal boom-bust cycles.5 For instance, Beaudry et al. [11, 13] show the existence of a recurrent peak

in several spectral densities of US trendless macroeconomic data suggesting the presence of periodicities around 9 to 10 years irrespective of the exogenous cyclical forces. At least their results run counter the empirical irrelevance of endogenous fluctuations.6 We build on these results to reconcile the predictions of our model with empirical evidence.

In this paper, we view the existence of such limit cycles for the inheritance flow (in level or a as share of national income) as a complementary interpretation of Piketty [51] and thus model and rationalize it by formally characterizing the corresponding complex dynamics, and providing some empirical evidence.

Capitalizing on Michel and Venditti [43], we do so through the lens of a two-sector overlapping generation (henceforth, OLG) model with a pure consumption good and one capital good, and a constant population of finitely-lived agents. We proceed in three steps, starting successively with the central planner problem and the optimal growth solution in the absence of legs. In the third and last step, we consider the decentralized problem where altruistic agents must determine their life-cycle consumptions, savings and inheritance to their children.7

The central planner problem and the corresponding growth solution lead to a dimension-four dynamical system and allow us to first characterize the degree-4 characteristic poly-nomial associated to the linearized dynamical system around the steady state. Building on the quasi-palindromic nature of the characteristic equation, we solve it and provide a complete assessment of its characteristic roots.8 In particular, we show that at least two

4For further evidence, see Section 6.

5Comin and Gertler [22] first provide evidence of medium-term cycles (with a periodicity between 8

and 50 years). See also Correa-L´opez et al. [23] for an application to medium-term technology cycles.

6In the same vein, Growiec et al. [33] conclude that the labor’s share of GDP exhibits medium-run

swings. See also Charpe et al. [19].

7As shown by Weil [59], when bequests are strictly positive across generations, the solution of the

Barro model is equivalent to the solution of a Ramsey-type optimal growth model where a central planner maximizes the total intertemporal welfare.

8If P (x) = P4

i=0

aixi is a polynomial of degree 4 and ai = an−i for i = 0, · · · , 2, then P is palindromic

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roots or a pair of complex conjugate roots have a modulus larger than one. To the best of our knowledge, it is the first available proposition that exploits the quasi-palindromic property of the characteristic equation in the literature of macroeconomic dynamic mod-els. This result is the cornerstone of our paper. First, it implies that the steady state can be either saddle-point stable or (totally) unstable. In the latter case, endogenous cycles can occur. Second, it is well-known that the existence of a Hopf bifurcation in mod-els featuring forward-looking agents and a competitive equilibrium structure requires the consideration of at least three sectors and thus of dimension-four dynamical systems. And as shown in several contributions, e.g. Magill [38, 39, 40] and Magill and Scheinkman [41], the curse of dimensionality prevents the derivation of meaningful sufficient conditions for the existence of complex characteristic roots. Our paper shows that only two sectors are sufficient in an OLG economy and it makes one step further to a better understanding of the occurrence of complex roots. We provide indeed clear-cut sufficient conditions for the existence of complex roots leading to a Hopf bifurcation, and as far as we know, this is the first time such conditions are exhibited in the literature. Third, we can identify two key mechanisms that lead to quasi-periodic cycles through a Hopf bifurcation as long as agents are sufficiently impatient. The first one is based on the properties of preferences, and especially the sign of the cross-derivative of the utility function or equivalently the elasticity of intertemporal substitution. The second rests on sectoral technologies through the sign of the capital intensity difference across sectors.9

Furthermore, these preference and technology-based mechanisms can either generate endogenous fluctuations independently or self-sustain themselves and thus amplify or mitigate (long-run) limit cycles. In the case of non-strictly concave preferences, mild perturbations of either the capital intensity difference across sectors or of the elasticity of intertemporal substitution lead to a flip bifurcation and thus to persistent period-2 cycles.10 The global dynamics can then be described as the product of two cycles

implying complex properties of the optimal path.11 On the one hand, the elasticity of

x4 m2P

m

x and P is quasi-palindromic. Importantly, using the change of variables z = x + m

x in P (x)

x2

produces a quadratic equation.

9The capital intensity difference across sectors has long been identified as a key driver of the dynamic

properties of two-sector optimal growth models.

10The consumption good is assumed to be more capital intensive than the investment good (see

Ben-habib and Nishimura [15]).

11The quasi-palindromic polynomial can be factored as the product of two order-2 polynomials where

one quadratic polynomial captures only the preference-based mechanism and the other only that based on technology.

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intertemporal substitution must be large enough to allow a substitution effect between the first and second period consumptions while satisfying the standard transversality conditions and the convergence towards the period-two cycle, and, on the other hand, the consumption good sector needs to be capital intensive to generate fluctuations of the capital stock.

In contrast, these two mechanisms can no longer be separated in the case of strictly concave preferences, and a Hopf bifurcation can occur with quasi-periodic cycles. On top of the empirical implications discussed later on, this result is again drastically dif-ferent from those in standard optimal growth models in which the existence of complex roots does require at least three sectors (e.g., Benhabib and Nishimura [15]). This can be explained by the fact that standard optimal growth models can only rely on the technology-based mechanism and higher-order dynamical systems (i.e., more sectors in the economy) are needed for quasi-periodic fluctuations. Here a two-sector optimal OLG growth problem with non-separable and strictly concave preferences is able to generate a Hopf bifurcation and quasi-periodic cycles taking some intermediate and plausible values for the elasticity of current consumption, the elasticity of intertemporal substitution and a upper threshold condition for the sectoral elasticities of capital-labor substitution.12

Turning to the decentralized problem in the presence of altruistic parents, we show that the optimal conditions and thus the dynamical system characterizing the decen-tralized equilibrium are equivalent to those associated with the central planner problem described above as long as bequests are strictly positive. After providing conditions for the existence of strictly stationary bequests, we argue that the (sufficient) conditions for optimal periodic and quasi-periodic cycles derived for the central planner solution also hold in the presence of positive bequests. This sharply contrasts with the predictions of the standard Barro model in which the optimal path monotonically converges toward the steady state if the life-cycle utility function of a representative generation living over two periods is additively separable. Also our result goes further than that of Michel and Venditti [43] in a one-sector model—endogenous period two-cycles can occur if the life-cycle utility function is non-additively separable with a positive cross derivative across periods. Finally, we show that both bequests and bequests as a share of GDP can be

12Kalra [36] and Reichlin [54] provide conditions for the existence of Hopf cycles in two-sector OLG

models but do not take into account bequests. See also Ghiglino and Tvede [30] for the analysis of endogenous cycles in general OLG models, and Ghiglino [29] for the analysis of the link between wealth inequality and endogenous fluctuations in a two-sector model.

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characterized by optimal periodic and quasi-periodic cycles. This result turns out to be insightful for our empirical results.

Looking at long-date inheritance flows data for France (Piketty, [51]), we finally pro-vide a quantitative assessment of the long-run cyclical behavior of bequests as a share of national income. In so doing, we proceed in two steps. First, using the low-frequency methodology of M¨uller and Watson ([47], [48]) and a band pass filter at low frequencies, we extract the long-run component of the inheritance variable of interest and then we discuss some statistical features of the remaining cyclical component. Notably, we pro-vide confidence intervals for the long-run standard deviation (variability) and contrast the filtered series with both a local-to-unit autoregressive model and a fractional model to further assess the persistence and (weakly) stationary properties. Second, following Beaudry et al. [11, 13], we make use of the spectral density to identify either a peak at a given frequency or a peak range over a low frequency interval, and thus to provide some support of recurrent (medium and long-run) cyclical fluctuations at that frequency (interval). Moreover, we test the presence of a shape restriction on the spectral density, i.e. the statistical significance of the ”peak range” of the spectral density at low frequency interval with respect to a flat prior. Our results strongly support the view of medium-term business fluctuations with periodicity between 24 and 40 years and the fact that the corresponding peak range is statistically significant. While the presence of a peak range does not necessarily imply strong endogenous cyclical forces and the empirical relevance of a Hopf bifurcation, it says again that data can not, at least, contradict the existence of endogenous (stochastic) limit cycles.13

The paper is organized as follows. Section 2 presents the two-sector model with non-additively separable preferences, defines the optimal growth problem of the central planner, proves the existence of a steady state, and derives the characteristic polynomial from which the stability analysis is conducted. The existence of period-two cycles under the assumption of a non-strictly concave utility function is discussed in Section 3 together with the presentation of a simple example to illustrate the main conditions. Section 4 contains the extension to the case of a strictly concave utility function. We provide general sufficient conditions that rule out the existence of complex characteristic roots and we consider a specific class of utility functions to prove the possible existence of a Hopf

13See Dufourt et al. [26] where the Hopf bifurcation is also shown to be relevant from an empirical

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bifurcation and thus of quasi-periodic cycles. In Section 5 we show that all our previous conditions are compatible with the decentralized equilibrium characterized by strictly positive bequests. Section 6 discusses our empirical results regarding the inheritance flows as a share of national using long-dated French annual data. Concluding comments are provided in Section 7 and all the proofs are contained into a final Appendix.

2

The model

2.1

Production

We consider a two-sector economy with one pure consumption good y0 and one capital

good y. Each good is produced with a standard constant returns to scale technology:

y0 = f0(k0, l0), y = f1(k1, l1)

with k0+ k1 ≤ k, k being the total stock of capital, and l0+ l1 ≤ 1, the total amount of

labor being normalized to 1.

Assumption 1. Each production function fi : R2+ → R+, i = 0, 1, is C2, increasing

in each argument, concave, homogeneous of degree one and such that for any x > 0, fkii(0, x) = flii(x, 0) = +∞, fkii(+∞, x) = flii(x, +∞) = 0.

For any given (k, y), we define a temporary equilibrium by solving the following prob-lem of optimal allocation of factors between the two sectors:

T (k, y) = max k0,k1,l0,l1 f0(k 0, l0) s.t. y ≤ f1(k1, l1) k0+ k1 ≤ k l0+ l1 ≤ 1 k0, k1, l0, l1 ≥ 0. (1)

The value function T (k, y) is called the social production function and describes the frontier of the production possibility set. Constant returns to scale of technologies imply that T (k, y) is concave non strictly. We will assume in the following that T (k, y) is at least C2.14

14A proof of the differentiability of T (k, y) under Assumption 1 and non-joint production is provided

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Let p denote the price of the investment good, r the rental rate of capital and w the wage rate, all in terms of the price of the consumption good, it is straightforward to show that:

Tk(k, y, ) = r(k, y), Ty(k, y) = −p(k, y) and w(k, y) = T (k, y) − r(k, y)k + p(k, y)y. (2)

We can also characterize the second derivatives of T (k, y). Using the concavity prop-erty we have:

Tkk(k, y) = ∂k∂r ≤ 0, Tyy(k, y) = −∂p∂y ≤ 0.

As shown by Benhabib and Nishimura [16], the sign of the cross derivative Tky(k, y)

is given by the sign of the relative capital intensity difference between the two sectors. Denoting a00 = l0/y0, a10 = k0/y0, a01 = l1/y and a11 = k1/y the capital and labor

coefficients in each sector, it is easy to derive from the constant returns to scale property that: dp dr = a01  a11 a01 − a10 a00  ≡ b (3)

with b the relative capital intensity difference, and thus Tky = Tyk = −∂p∂r∂r∂k = −Tkkb.

The sign of both b and Tky is positive if and only if the investment good is capital intensive.

Note also that Tyy(k, y) can be written as:

Tyy = −∂p∂r∂y∂r = Tkkb2.

Remark 1 : The derivative dr/dp = b−1 is well-known in trade theory as the Stolper-Samuelson effect. Similarly, at constant prices, we can derive the associated Rybczinsky effect dy/dk = b−1. We therefore find the well-known duality between the Rybczinsky and Stolper-Samuelson effects.

2.2

Preferences

The economy is populated by a constant population of finitely-lived agents.15 In each

period t, Nt = N persons are born, and they live for two periods: they work during the

first (with one unit of labor supplied) and they have preferences for consumption (ct,

when they are young, and dt+1, when they are old) which are summarized by the utility

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function u(ct, Bdt+1), with B > 0 a normalization constant, such that Assumption 2 is

satisfied.

Assumption 2. u(c, Bd) is increasing in both arguments (uc(c, Bd) > 0 and ud(c, Bd) >

0), concave and C2 over the interior of R2

+. Moreover, limX→0XuX(c, X)/uc(c, X) = 0

and limX→+∞XuX(c, X)/uc(c, X) = +∞, or limX→0XuX(c, X)/uc(c, X) = +∞ and

limX→+∞XuX(c, X)/uc(C, X) = 0.

We also introduce a standard normality assumption between the two consumption levels.

Assumption 3. Consumptions c and d are normal goods.

We finally consider the following useful elasticities of consumptions:

cc = −uc/uccc > 0, cd = −uc/ucdBd, (4)

dc = −ud/ucdc, dd = −ud/uddBd > 0 (5)

It is worth noting that the normality Assumption 3 implies 1/cc − 1/dc ≥ 0 and

1/dd − 1/cd ≥ 0 and concavity in Assumption 2 implies 1/(ccdd) − 1/(dccd) ≥ 0.

Taking these elasticities, the elasticity of intertemporal substitution between ct and dt+1

writes: ς(ct, dt+1) = ud(ct,dt+1)/uc(ct,dt+1) ct/dt+1 ∂(ud(ct,dt+1)/uc(ct,dt+1)) ∂(ct/dt+1) = 1 1 cc− 1 dc ≥ 0. (6)

This elasticity will be used as a parameter driving the existence of endogenous fluctua-tions.

2.3

The optimal growth problem

Under complete depreciation within one period,16 the capital accumulation equation is:

kt+1 = yt. (7)

Since total labor is normalized to 1, we consider from now on that N = 1. At each time t, total consumption is then given by the social production function, i.e. ct+ dt = T (kt, yt).

The intertemporal objective function of the central planner combines utilities of successive generations:

16Considering that in an OLG model one period is approximately 30 years, complete depreciation is a

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max {ct,dt+1} +∞ X t=0 βtu(ct, Bdt+1) (8)

where β ∈ (0, 1] is the discount factor.17 Considering (7) and the fact that c

t= T (kt, yt)−

dt, the optimization program (8) can be equivalently written as follows

max {dt+1,kt+1} +∞ X t=0 βtu(T (kt, kt+1) − dt, Bdt+1) (9)

with d0 and k0 given. The first order conditions are given by the following two difference

equations of order two:

ud(T (kt, kt+1) − dt, Bdt+1)B − βuc(T (kt+1, kt+2) − dt+1, Bdt+2) = 0

uc(T (kt, kt+1) − dt, Bdt+1)Ty(kt, kt+1) +

βuc(T (kt+1, kt+2) − dt+1, Bdt+2)Tk(kt+1, kt+2) = 0.

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Notably, taking some (given) initial conditions (d0, k0), any path that satisfies equations

(10) together with the following transversality conditions, lim t→+∞β tu d(ct, Bdt+1)pt+1kt+1 = 0 and lim t→+∞β tu d(ct, Bdt+1)dt+1= 0, (11) is an optimal path.

2.4

Steady state

A steady state is defined as the stationary solution, kt = k∗, dt = d∗, for all t of the

following nonlinear system of equations:

ud(T (k,k)−d,Bd)B

uc(T (k,k)−d,Bd) = β

−Ty(k,k)

Tk(k,k) = β.

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Beside discussing the existence and uniqueness of the steady state, note that we further introduce the B parameter to normalize the stationary consumption d such that it remains constant when the discount factor β is modified. As in the standard two-sector model, we get the following result:

Proposition 1. Under Assumptions 1-3, there exists a unique steady state (k∗, d∗) solu-tion of equasolu-tions (12). Moreover, there exists a unique value B∗ such when B = B∗, the stationary consumption d∗ can be normalized to any value ¯d ∈ (0, T (k∗, k∗)).

17In the case β = 1, the infinite sum into the optimization program (8) may not converge. In such a

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Proof. See Appendix 8.1.

A pair (k∗, d∗) is then defined to be the Modified Golden Rule. Finally, the stationary consumption of young agents is obtained from c∗ = T (k∗, k∗) − d∗.

2.5

Characteristic polynomial

We are now in a position to derive the characteristic polynomial from total differentiation of equations (10). Denoting T (k∗, k∗) = T∗, Tk(k∗, k∗) = Tk∗ and Tkk(k∗, k∗) = Tkk∗ , the

elasticities of the consumption good’s output and the rental rate with respect to the capital stock, all evaluated at the steady state, can be written as:

εck = Tk∗k ∗

/T∗ > 0, εrk = −Tkk∗ k ∗

/Tk∗ > 0. (13)

Then Proposition 2 yields the degree-4 characteristic polynomial and discusses the multiplicity order of the possible (characteristic) roots.

Proposition 2. Under Assumptions 1-3, the degree-4 characteristic polynomial is given by P(λ) = λ4− λ3B + λ2C − λB β + 1 β2 (14) with B = −bβ cc εck εrk  cc dc − cd dd  + β+bβb2 + dc βcc + cd dd C = −(1+β)b cc εck εrk  cc dc − cd dd  +β+bβb2 dc βcc + cd dd  +β2 (15) or equivalently P(λ) =hλ2− λdc βcc + cd dd  + 1β i (λb−1)(λβ−b) βb + λ(λ − 1)  λ − 1β  β bcc εck εrk  cc dc − cd dd  (16)

If λ is a characteristic root of (16), then ¯λ, (βλ)−1 and (β ¯λ)−1 are also characteristic roots of (16). Moreover, at least two roots or a pair of complex conjugate roots have a modulus larger than one, and one of the following cases necessarily hold:

i) the four roots are real and distincts,

ii) the four roots are given by two pairs of non-real complex conjugates, iii) there are two complex conjugates double roots or two real double roots.

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Proposition 2 is of critical importance and several points are worth commenting. First, it shows that if there exists a pair of complex characteristic roots (λ, ¯λ) solutions of the quartic polynomial (16), then a second pair of complex characteristic roots, (βλ)−1 and (β ¯λ)−1, are also solutions of (16). Therefore, Proposition 2 proves that the 4 characteristic roots are either all real or all complex. Second, Proposition 2 also implies that at most two characteristic roots can have a modulus lower than 1 and thus that the steady state can be either saddle-point stable or totally unstable. Of course in this last case, endogenous cycles can occur. Third, under Assumption 2, the sign of the expression cc

dc − cd

dd is given

by the sign of the cross derivative ucd, i.e. by the opposite of the sign of cd, dc, which is a

crucial ingredient to determine the local stability properties of the steady state. Fourth, using Eq. (16), when the utility function is non-strictly concave, i.e. if cc

dc − cd dd = 0,

then the degree-4 polynomial simplifies to a product of two degree-2 polynomials. In this respect, we first focus on the simpler case of a non-strictly concave utility function (Section 4), and then consider the more general case of strictly concave preferences (Section 5).

Remark 2: The degree-4 characteristic polynomial (14) is a quasi-palindromic equa-tion that can be solved explicitly, and its roots can be determined using only quadratic equations (see Appendix 8.9 for details.). As far as we know, this is the first time this type of methodology is applied to macroeconomic dynamic models. The existence of a Hopf bifurcation in models featuring forward-looking agents and a competitive equilib-rium structure has initially been emphasized in Benhabib and Nishimura [15]. However, due to the dual structure of such models, the consideration of at least three sectors and of dimension-four dynamical systems has been shown to be necessary to get such determin-istic fluctuations based on the existence of complex characterdetermin-istic roots. And as shown in many contributions, i.e. Magill [38, 39, 40] and Magill and Scheinkman [41], due to the high dimension of such systems, there is no obvious sufficient conditions for the existence of complex characteristic roots. In this paper, using the quasi-palindromic structure of the characteristic polynomial allows us to provide such sufficient conditions.

Remark 3: If b = 0, one gets the one-sector formulation with a two-dimensional dy-namical system as considered in Michel and Venditti [43]. Indeed, the characteristic polynomial can be simplified as follows

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P(λ) = λ2− λβccdc +cddd− (1+β) cc εckεrk  cc dc−ddcd  1−ccβ εck εrk  cc dc− cd dd  + 1 β

The same conclusions as in Michel and Venditti [43] are obviously derived.

Similarly, if the utility function is additively separable, i.e. ucd = udc = 0, we get

the two-sector optimal growth formulation with a two-dimensional dynamical system as considered in Benhabib and Nishimura [15]. The characteristic polynomial can indeed be simplified as follows P(λ) = λ2− λ(1 + β) β cc εck εrk+(β+b 2) β cc εck εrk+(1+β)b + 1 β

The same conclusions as in Benhabib and Nishimura [15] are then derived.

3

Period-two cycles under non-strictly concave

pref-erences

In this section we assume that the utility function is non-strictly concave.

Assumption 4. The utility function u(c, Bd) is concave non-strictly, i.e. cc dc −

cd dd = 0.

In so doing, one can show that the characteristic roots cannot be complex.

Lemma 1. Under Assumptions 1-4, the characteristic roots are real.

Proof. See Appendix 8.3.

Following simultaneously the same methodologies as in the two-sector optimal growth model and the optimal growth solution of the aggregate OLG model, we discuss the local stability properties of equilibrium paths depending both on the sign of the capital intensity difference across sectors b and the sign of the cross derivative ucd, i.e. of the two

elasticities cd and dc.

As a first step, Proposition 3 provides some simple conditions ensuring the saddle-point property with monotone convergence.

Proposition 3. Under Assumptions 1-4, if b ≥ 0 and cd, dc ≥ 0, i.e. ucd ≤ 0, then the

equilibrium path is monotone and the steady-state (k∗, d∗) is a saddle-point.

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3.1

A separated mechanism for period-two cycles

We now show that convergence with oscillations and persistent competitive equilibrium cycles may occur under a quite large set of circumstances.

Proposition 4. Under Assumptions 1-4, the following results hold:

i) When the investment good is capital intensive, i.e. b ≥ 0, let cd, dc < 0, i.e.

ucd > 0. Then the steady state (k∗, d∗) is saddle-point stable with damped oscillations if

and only if the elasticity of intertemporal substitution satisfies ς ∈ (0, ς) ∪ (¯ς, +∞) with ς ≡ ccβ

1+β and ¯ς ≡ cc

2

Moreover, when ς crosses the bifurcation values ς or ¯ς, (k∗, d∗) undergoes a flip bifurcation leading to persistent period-2 cycles.

ii) When cd, dc ≥ 0, i.e. ucd ≤ 0, let the consumption good be capital intensive, i.e.

b < 0. Then the steady state (k∗, d∗) is saddle-point stable with damped oscillations if and only if b ∈ (−∞, −1) ∪ (−β, 0). Moreover, if there is some β∗ ∈ (0, 1) such that b ∈ (−1, −β∗), then there exists ¯β ∈ (0, 1) such that, when β crosses ¯β from above, (k∗, d∗) undergoes a flip bifurcation leading to persistent period-2 cycles.

iii) When the consumption good is capital intensive, i.e. b < 0, and cd, dc < 0, i.e.

ucd > 0, the steady state (k∗, d∗) is saddle-point stable with damped oscillations if and

only if b ∈ (−∞, −1) ∪ (−β, 0) and ς ∈ (0, ς) ∪ (¯ς, +∞). Moreover, if there is some β∗ ∈ (0, 1) such that b ∈ (−1, −β∗), then there exists ¯β ∈ (0, 1) such that, when β crosses

¯

β from above or ς crosses the bifurcation values ς or ¯ς, (k∗, d∗) undergoes a flip bifurcation leading to persistent period-2 cycles.

Proof. See Appendix 8.5.

Proposition 4 provides two independent mechanisms leading to the existence of endoge-nous fluctuations. The first one is based on the properties of preferences through the sign of the cross derivative ucd and is the most interesting in our context since it allows to

generate period-2 cycles in a two-sector model even under a capital intensive investment good sector—a condition which is known since Benhabib and Nishimura [16] to ensure monotone convergence in a standard two-sector optimal growth model.

To provide further economic insights, let us consider an instantaneous increase in the capital stock kt. Using ct+dt = T (kt, yt) and Tk> 0, it follows that ctincreases, and thus,

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taking that the marginal utility of second period consumption ud is larger as udc > 0, a

constant utility level u(ct, dt+1) can be obtained from a decrease of dt+1. Using the first

equation in (10) and taking dt+1 as given, ∆ct+1 ∆ct = udc uccβ + udd uccβ ∆dt+1 ∆ct < 0.

Finally, since ct+1+dt+1= T (kt+1, yt+1), total consumption at time t+1 is lower, which in

turn implies that a lower capital stock kt+1 when yt+1 holds constant. Endogenous

fluc-tuations are thus generated from intertemporal consumption allocations based on some substitution effects between the first and second period consumptions. The important result is that the elasticity of intertemporal substitution needs to be large enough to allow sufficient substitution between ct and dt+1 to generate aggregate oscillations, but

should not be too large to be compatible with the transversality conditions (11) and a convergence process towards the period-two cycle.

The second mechanism is, as in the two-sector optimal growth model, based on the properties of sectoral technologies through the sign of the capital intensity difference across sectors. Following Benhabib and Nishimura [16], we can use the Rybczinski and Stolper-Samuelson effects to provide a simple economic intuition for this result. Assume indeed that the consumption good is capital intensive, i.e. b < 0, and consider an instantaneous increase in the capital stock kt. This results in two opposing mechanisms:

- On the one hand, the trade-off in production becomes more favorable to the consump-tion good, and the Rybczinsky effect implies a decrease of the output of the capital good yt. This tends to lower both the investment and the capital stock in the next period kt+1.

- On the other hand, in the next period the decrease of kt+1 implies again through

the Rybczinsky effect an increase of the output of the capital good yt+1. Indeed the

decrease of kt+1 improves the trade-off in production in favor of the investment good

which is relatively less intensive in capital and this tends to increase the investment and the capital stock in period t + 2, kt+2.

Of course, under both mechanisms, the existence of persistent fluctuations require that the oscillations in consumption and relative prices must not present intertemporal arbitrage opportunities. Consequently, a minimum level of myopia, i.e. a low enough value for the discount rate β, is thus necessary.

Note finally that in case iii) of Proposition 4, both mechanisms hold at the same time. Interestingly, using both β and ς as two bifurcation parameters allows to consider a

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co-dimension 2 bifurcation which corresponds to the flip bifurcation with a 1:2 resonance where two characteristic roots are equal to −1 simultaneously. As shown in Kuznetsov [37], in such a configuration, under non-degeneracy conditions, the steady state is either saddle-point stable or elliptic. This last case may give rise to the existence of quasi-periodic cycles which are usually associated to a Hopf bifurcation.

3.2

A simple illustration with homogeneous preferences

To shed more light on Proposition 4, we provide an example for all cases. In doing so, we consider the class of homogeneous of degree γ ≤ 1 utility functions with B = 1,18 which obviously satisfies Assumptions 2 and 3. Then we introduce the share of first period consumption within total utility φ(c, d) ∈ (0, γ) defined by:

φ(c, Bd) = uc(c,Bd)c

u(c,Bd) . (17)

Accordingly, the share of second period consumption within total utility is defined as γ − φ(c, Bd) ∈ (0, 1). Consequently, using (4)-(5), the elasticities of interest are given by

cd = −1−cccc(1−γ), dc = −φ[1−(γ−φ)cc(1−γ)]cc , dd = φ−cc(γ−φ)(1−γ)(2φ−γ)cc . (18)

Furthermore, we impose a restriction on cc to ensure concavity and the normality goods

assumption.

Assumption 5. cc ≤ φ(1−γ)γ ≡ ¯cc

Therefore it is straightforward to get dd > 0 while cd, cd < 0 if and only if cc <

1/(1 − γ) ≡ ˜cc(< γ/φ(1 − γ)). Moreover, the elasticity of substitution between the two

life-cycle consumption levels is now defined by: ς(φ) = cc(γ−φ)

γ−φcc(1−γ) ∈ (0, +∞) (19)

Notably, if cc < ˜cc, then ς(φ) ∈ (0, cc).

Finally, from the production side, as in Baierl et al. [8], we assume that the con-sumption and investment goods are produced with Cobb-Douglas technologies as follows

y0 = k0α0l 1−α0 0 , y = k α1 1 l 1−α1 1 (20)

It can be shown that b = β(α1−α0)

1−α0 (21)

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Finally, for sake of simplicity, γ is set to one, which means that Assumptions 4 and 5 hold, and that ς = cc(1 − φ) < cc.

We first discuss case iii) of Proposition 4 where a co-dimension 2 bifurcation can arise. In that respect, we derive the following Corollary:19

Corollary 1. Let the utility function be homogeneous of degree 1 and the sectoral produc-tion funcproduc-tions be given by (20), and assume that α0 > (1 + α1)/2. Then the steady state

(k∗, d∗) is saddle-point stable with damped oscillations if and only if ς ∈ (0, ς) ∪ (¯ς, cc)

and β > β, with ς = ccβ 1+β, ¯ς = cc 2 and β = 1−α0 α0−α1

If β = β and ς = ¯ς or ς, then a co-dimension 2 flip bifurcation with a 1:2 resonance generically occurs.

Proof. See Appendix 8.6.

While providing a precise dynamic analysis of this co-dimension 2 bifurcation goes far beyond the objectives of this paper, it is worthwhile to mention that this case provides an interesting possibility of smooth endogenous fluctuations for the main aggregate variables which does not arise under a standard flip bifurcation. Indeed, while there does not a priori exist complex characteristic roots under a linear homogenous utility function, Kuznetsov [37] shows that under a 1:2 resonance, the steady state can be elliptic and a stable limit cycle, similar to those that arise under a Hopf bifurcation, can occur. As pointed out in the next section, a Hopf bifurcation provides another tool to describe the long-run cyclical behavior of macroeconomic variables such as bequests.

Second, we focus on case (i) of Proposition 4 in which the investment good is capital intensive. The occurrence of endogenous fluctuations is then characterized in Corollary 2. Note that the result does not depend on the sign of the capital intensity difference and also holds if the consumption good is capital intensive.

Corollary 2. Let the utility function be homogeneous of degree 1 and the sectoral produc-tion funcproduc-tions be given by (20). Then, for any of α0, α1 ∈ (0, 1), the steady state (k∗, d∗)

is saddle-point stable with damped oscillations if and only if ς ∈ (0, ς) ∪ (¯ς, cc). Moreover,

19Similar conclusions can be derived using instead CES technologies with non unitary sectoral

elas-ticities of capital-labor substitution. It can be shown indeed that our conclusions are robust to a wide range of values for these parameters. A proof of this claim is available upon request.

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when ς crosses the bifurcation values ς or ¯ς, (k∗, d∗) undergoes a flip bifurcation leading to persistent period-2 cycles.

Finally, we focus on case (ii) of Proposition 4 where endogenous fluctuations arise for any given value ς ∈ (0, ς) ∪ (¯ς, cc) when the consumption good sector is capital intensive.

Corollary 3. Let the utility function be homogeneous of degree 1 and the sectoral produc-tion funcproduc-tions be given by (20). Assume also that ς ∈ (0, ς) ∪ (¯ς, cc) and α0 > (1 + α1)/2.

Then the steady state (k∗, d∗) is saddle-point stable with damped oscillations if and only if β > β. Moreover, when β crosses the bifurcation value β, (k∗, d∗) undergoes a flip bifurcation leading to persistent period-2 cycles.

This case is presented here for illustration as it corresponds to the main conclusions of Benhabib and Nishimura [16] derived in a standard two-sector optimal growth model.

4

Quasi-periodic cycles under strictly concave

pref-erences

As explained above, using a non-strictly concave utility function is convenient in the sense that it reduces the 4 characteristic polynomial to the product of two degree-2 polynomials. In such a framework, we have shown that the characteristic roots are necessarily real and that endogenous fluctuations can occur through the existence of period-two cycles. Notably the preference and technology mechanisms are separated and lead independently to the occurrence of endogenous fluctuations. Relaxing this sim-plifying assumption, our objective here is to prove that the preference and technology mechanisms can mix together, and then complexify and amplify the possible endogenous fluctuations in the context of strictly concave preferences.

4.1

A mixed mechanism for the existence of quasi-periodic

cy-cles

We now need to focus on the existence of complex characteristic roots and quasi-periodic cycles occurring through a Hopf bifurcation. Therefore we start by providing general sufficient conditions allowing to rule out the existence of complex roots.

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Proposition 5. Under Assumptions 1-3, let the utility function u(c, Bd) be strictly con-cave. Then the roots of the characteristic polynomial (16) are necessarily real in the following cases:

i) for any sign of cd, cd if the investment good sector is capital intensive, i.e. b > 0,

ii) if cd, cd > 0 and the consumption good sector is capital intensive, i.e. b < 0.

Proof. See Appendix 8.7.

Necessary conditions for the existence of complex roots are therefore based on the two mechanisms that generate endogenous fluctuations in the non-strictly concave case, namely b < 0 and cd, cd< 0.

In order to study whether complex characteristic roots and a Hopf bifurcation with quasi-periodic cycles can occur, we consider again a utility function homogeneous of degree γ, but with γ < 1 to allow for strict concavity.

We first derive sufficient conditions to ensure saddle-point property of the steady state with real characteristic roots.

Proposition 6. Let the utility function be homogeneous of degree γ < 1, and assume that cc < ˜cc, b ∈ (−∞, −1) ∪ (−β, 0) and

−εck

bεrk > 1 (22)

Then there exist 0 < ς ≤ ¯ς < cc and ˆcc ∈ (0, ˜cc) such that when ς ∈ (0, ς) ∪ (¯ς, cc)

the characteristic roots are real and the steady-state is saddle-point stable. Moreover, i) when ς ∈ (0, ς), the optimal path converges towards the steady state with oscillations if cc ∈ (0, ˆcc) or monotonically if cc ∈ (ˆcc, ˜cc),

ii) when ς ∈ (¯ς, cc), the optimal path converges towards the steady state with

oscilla-tions.

Proof. See Appendix 8.8.

Condition (22) allows to get the existence of the bound ˆcc and thus the occurrence

of oscillations when the elasticity of intertemporal substitution is low, i.e. ς ∈ (0, ς). This restriction can be easily interpreted. Denoting σi the elasticity of capital-labor

substitution in sector i = 0, 1 and using Drugeon [25], we can relate the ratio of elasticities εck/εrk to an aggregate elasticity of substitution between capital and labor, denoted Σ,

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which is obtained as a weighted sum of the sectoral elasticities σi:20 εck εrk =  T l2 0  s 1−s Σ GDP with Σ = GDP pykT (pyk0l0σ0+ T k1l1σ1) , (23)

GDP = T + py and s = rk/GDP the share of capital income in GDP. Therefore, os-cillations when φ ∈ ( ¯φ, γ) are associated with a large aggregate elasticity of substitution between capital and labor i.e., large enough sectoral elasticities of capital-labor substitu-tion.

Proposition 6 implies that the existence of complex roots, if any, requires to consider intermediate values for the elasticity of intertemporal substitution ς between the first and second period consumptions, i.e. ς ∈ (ς, ¯ς). As mentioned previously, the degree-4 char-acteristic polynomial (14) is a quasi-palindromic equation that can be solved explicitly, and its roots can be determined using only quadratic equations. As shown in Appendix 8.9, we apply this methodology in order to provide sufficient conditions for the occurrence of complex roots and a Hopf bifurcation. As far as we know, this is the first time this type of sufficient conditions are exhibited in 4-dimensional optimal growth models. We there-fore provide some positive issue to the previous contributions, i.e. Magill [38, 39, 40] and Magill and Scheinkman [41], where it has been shown that no obvious sufficient conditions for the existence of complex characteristic roots can be exhibited.

We can indeed derive the following result:

Proposition 7. Let the utility function be homogeneous of degree γ < 1, and assume that cc < ˜cc and b ∈ (−β, 0). Then there exist ¯b ∈ (−β, 1), γ ∈ (0, 1), cc, ¯cc ∈ (0, ˜cc),

¯

ε > 0 and four critical values (ς ≤)ςc < ςH < ¯ςH < ¯ςc(≤ ¯ς) such that when b ∈ (−β, ¯b),

γ ∈ (γ, 1), cc ∈ (cc, ¯cc) and

−εck

bεrk < ¯ε (24)

the following results hold:

i) the steady state (k∗, d∗) is saddle-point stable with damped oscillations if ς ∈ (ςc, ςH) ∪ (¯ςH, ¯ςc),

ii) when ς crosses the bifurcation values ςH or ¯ςH, (k, d) undergoes a Hopf bifurcation

leading to persistent quasi-periodic cycles.

Proof. See Appendix 8.9.

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From a theoretical point of view, Proposition 7 provides a strong conclusion as it shows that a Hopf bifurcation and quasi-periodic cycles can occur in a two-sector optimal growth framework as long as it is based on an OLG structure with non-separable and strictly concave preferences. More specifically, we need intermediate values for the elasticity cc

and the elasticity of intertemporal substitution ς between the first and second period consumptions, together with, using (23), a not too large value for the sectoral elasticities of capital-labor substitution.

Such a result is drastically different from what can be obtained in standard optimal growth models as the existence of complex roots requires to consider at least three sec-tors.21 This can be explained by the fact that we can mix here two mechanisms based

on both preferences and technology, while in standard optimal growth models only the technology mechanism arises and thus requires more than two sectors to generate quasi-periodic fluctuations. In our case, the preference mechanism based on a substitutability effect between first and second period consumptions, and the technology mechanism based on a capital intensive consumption good sector feed each other when the utility function is strictly concave and amplify the endogenous fluctuations of capital and con-sumption. More complex quasi-periodic fluctuations can thus occur for adequate values of the elasticity of intertemporal substitution in consumption.

4.2

A simple illustration

Let us now focus on a numerical illustration. Considering that the annual discount factor is often estimated to be around 0.96 and that one period in an OLG model is about 30 years, we consider here that β = 0.9630 ≈ 0.294. Focusing on a slight deviation with respect to the linear homogeneous case with γ = 0.98, let us then assume a standard value cc = 1 that satisfies cc ∈ (cc, ¯cc). We also consider sectoral Cobb-Douglas technologies

as given by (20) with α0 = 0.6 and α1 = 0.21 so that the consumption good is capital

intensive with b ≈ −0.28665 close to −β.22 The bounds exhibited in Proposition 7 are equal to ςc ≈ 0.1194 and ¯ςc ≈ 0.6083. We then find that the characteristic polynomial

(60) admits four characteristic roots λ1, λ2, λ3, λ4 that are complex conjugate by pair

21See Benhabib and Nishimura [15], Cartigny and Venditti [18], Venditti [58].

22The existence of a Hopf bifurcation can also be obtained using instead CES technologies with non

unitary sectoral elasticities of capital-labor substitution. As in the case with period-two cycles, our conclusions are robust to a wide range of values for these parameters. A proof of this claim is also available upon request.

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with λ1λ2 > 1 and λ3λ4 < 1 if ς ∈ (ς, ςH) ∪ (¯ςH, ¯ς) while λ3λ4 > 1 if ς ∈ (ςH, ¯ςH), with

ςH ≡ 0.3193 and ¯ςH ≡ 0.426. Moreover λ

3λ4 = 1 when ς = ςH or ¯ςH. As a result ςH and

¯

ςH are Hopf bifurcation values giving rise to quasi-periodic cycles in their neighborhood.

5

The solution with altruistic agents and a bequest

motive

As a final step, we need to show that the existence of optimal endogenous cycles is com-patible with strictly positive bequest transmissions across generations. We thus consider a decentralized economy composed of overlapping generations of parents loving their children. As in the Barro [9] formulation, each agent is altruistic towards his descendant through a bequest motive.23 Parents indeed care about their child’s welfare by taking into

account their child’s utility into their own utility function. They are now price-takers, considering as given the prices pt, wt and rt+1 as defined by (2), and determine their

optimal decisions with respect to their budget constraints

wt+ ptxt= ct+ ζt and Rt+1ζt= dt+1+ pt+1xt+1 (25)

with Rt+1 = rt+1/pt the gross rate of return, ζt the savings of young agents born in t

and xt the amount of bequest transmitted at time t by agents born in t − 1. Note that

bequest xt is expressed as an investment good and requires the relative price pt to enter

the budget constraints. In each period, bequests must be non-negative:

xt≥ 0 for all t ≥ 0 (26)

An altruistic agent has a utility function given by the following Bellman equation Vt(xt) = max {ct,dt+1,st,xt+1} {u(ct, Bdt+1) + βVt+1(xt+1)} = max {ct,dt+1,st,xt+1} +∞ X t=0 βtu(ct, Bdt+1) (27)

subject to (25) and (26). Note that β is now interpreted as the intergenerational degree of altruism.

23The co-existence of altruistic and non-altruistic agents as in Nourry and Venditti [50] could also be

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5.1

The decentralized equilibrium with positive bequests

It is well-known from the first welfare theorem that this altruistic problem is equivalent to the central planner problem (8), and the equilibrium is the unique Pareto optimum which coincides with the centralized solution. However, such an equivalence requires the non-negativity constraints of bequests (26) to hold with a strict inequality in order to preserve the link across generations. Substituting the expressions of ct and dt+ 1 from

the budget constraints (25) into the optimization problem (27) we get

max {ζt,xt+1} +∞ X t=0 βtu(wt+ ptxt− ζt, B[Rt+1ζt− pt+1xt+1]) (28)

The first order conditions are given by

uc(ct, Bdt+1) − ud(ct, Bdt+1)BRt+1 = 0 (29)

βuc(ct+1, Bdt+2) − ud(ct, Bdt+1)B ≤ 0 with an equality if xt> 0 (30)

Consider now the two budget constraints in (25) evaluated at the steady state. Solving with respect to ζt, and using the fact that ζt= ptyt= ptkt+1 and Rt+1= rt+1/pt, we get

p∗x∗ 1 −R1∗  = c∗+ Rd∗∗ − w∗ = T (k∗, k∗) − w∗− d∗ 1 −R1∗  = (r∗k∗− d∗) 1 − 1 R∗ . (31)

If x∗ > 0, i.e. r∗k∗ > d∗, then we derive from the first order conditions that R∗ = r∗/p∗ = β−1 and ud(c∗, Bd∗) = βuc(c∗, Bd∗), which are exactly the same conditions as (12). Then

Proposition 8. Under Assumptions 1-3, for any β ∈ (0, 1), there exists a unique value B∗ such that when B = B∗, bequests are positive in the economy with a degree of altruism equal to β.

Proof. See Appendix 8.10.

When bequests are positive at the steady state, then by continuity there are positive in a neighborhood of the steady state and we may study the local stability properties of the equilibrium path considering equation (30) with an equality. We need first to derive the precise expression of the dynamical system. Plugging equation (29) into equation (30) considered with an equality, and using the fact that Rt+1= rt+1/pt, we get the following

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uc(ct, Bdt+1)pt− βrt+1uc(ct+1, Bdt+2) = 0 (32)

βuc(ct+1, Bdt+2) − ud(ct, Bdt+1)B = 0. (33)

Taking now that pt = −Ty(kt, kt+1), rt+1 = Tk(kt+1, kt+2) and ct = T (kt, kt+1) − dt, we

immediately conclude that these equations are identical to the two difference equations of order two given by (10). Solving these two equations considering the transversality condi-tions (11) yields the equilibrium paths for capital {kt}t≥0and second period consump-tion

{dt}t≥0. The dynamics of bequests is then derived from the budget constraints (25) as

follows

ptxt = rtkt− dt (34)

whereas the dynamics of bequests as a proportion of GDP is

ptxt GDPt = ptxt T (kt,kt+1)+ptyt = s(kt, kt+1) − dt T (kt,kt+1)−Ty(kt,kt+1)kt+1. (35) with s(kt, kt+1) = T (kt,kt+1Tk)−T(kt,kyt+1(kt,k)kt+1t )kt+1

the share of capital income in GDP. We can then derive the following Proposition:

Proposition 9. Under Assumptions 1-3, the local stability properties provided in Proposi-tions 3, 4 and 7 hold for both bequests as defined by (34) and bequests as a proportion of GDP as defined by (35). In particular, bequests and bequests as a proportion of GDP can be characterized by optimal periodic and quasi-periodic cycles.

Proof. See Appendix 8.11.

This Proposition may explain long-run fluctuations of both bequests and bequests as a proportion of GDP, and provides a theoretical basis to explain the empirical evidence derived in Section 6 showing that the annual inheritance flow as a fraction of national income displays medium- and long-run fluctuations.

5.2

An illustration with positive bequests and endogenous

fluc-tuations

We consider again our benchmark example in Section 3.2. with a linear homogeneous utility function (γ = 1), and the Cobb-Douglas production structure described by (20). Using the expression of the elasticity of substitution ς as given by (19) and the expression

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for the stationary capital stock (56) in Appendix 8.6, we derive that r∗k∗ > d∗ and thus x∗ > 0 if and only if

α0β(cc−ς)

ς − (1 − α0− βα1) > 0

It follows immediately that if α1 > 1 − α0 and β > (1 − α0)/α1, then 1 − α0− βα1 < 0

and x∗ > 0 for any ς ∈ (0, cc). The existence of periodic cycles is thus compatible with

positive bequests. Similarly, when α1 < 1 − α0, straightforward computations show that

x∗ > 0 if and only if

ς < α0βcc

1−α0+β(α0−α1) ≡ ˜ς1

Therefore the conditions of Corollary 2 for the existence of period-2 cycles can be satisfied if ˜ς1 > ¯ς = cc/2. Sufficient conditions for this inequality to be satisfied are given by

α1 ∈ (1 − 2α0, 1 − α0) and β > (1 − α0)/(α0+ α1) ≡ β with β < 1. This example clearly

shows that when the degree of altruism is large enough, endogenous optimal fluctuations are compatible with positive bequests. Moreover, this result holds for any sign of the capital intensity difference across sectors.

It is worth noticing that if, under α1 ∈ (1 − 2α0, 1 − α0), we assume that ς < ˜ς1

with ˜ς1 < ς, then bequests are positive but the conditions of Corollary 2 for the existence

of period-2 cycles cannot be satisfied and the steady state is saddle-point stable. This inequality is satisfied if and only if α1 ∈ (1 − 2α0, 1 − α0), α0 < 1/2 and β < (1 − 2α0)/α1.

Therefore, if the degree of altruism is not large enough, persistent endogenous fluctuations cannot arise.

Let us finally illustrate the possible existence of quasi-periodic cycles under positive bequests when the utility function is homogeneous of degree γ < 1 as in Section 4. Using again the expression of the elasticity of substitution ς as given by (19) and the expression for the stationary capital stock (56) in Appendix 8.6, we derive that r∗k∗ > d∗ and thus x∗ > 0 if and only if

α0φβ − (γ − φ) (1 − α0− βα1) > 0

Consider then the particular illustration in Section 4 which is such that 1 − α0− βα1 > 0

and α0 > α1. It follows that bequests are positive if and only if

ς < α0βcc

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This condition can be satisfied only if

cc < (1−γ)(1−α1−α0+β(α0−α1) 0−βα1)

With γ = 0.98, cc = 1, α0 = 0.6 and α1 = 0.21, we get ˜ςγ ≈ 0.3566 ∈ (ςH, ¯ςH). It follows

that positive bequests are compatible with quasi-periodic cycles. Indeed, the steady state, which is characterized by strictly positive bequests if ς < ˜ςγ, is saddle-point stable with

damped oscillations if and only if ς ∈ (ςc, ςH). Moreover, when ς crosses the bifurcation

values ςH from below, the steady state undergoes a Hopf bifurcation leading to persistent quasi-periodic cycles and thus long-run fluctuations of bequests.

6

Long-run cycles and inheritance: an empirical

as-sessment

The objective of this section is to provide a qualitative assessment of the existence of limit cycles of bequests (relative to national income) in the light of our results in Section 5. Especially, we discuss indirect empirical evidence that is consistent with the emergence of a period-2 cycle or of (quasi-) periodic (long-run) cycles.24

6.1

Empirical relevance of flip versus Hopf bifurcations

One critical issue is that the dynamics engendered by either a flip or a Hopf bifurcation have different implications from an empirical point of view. The former leads to a period-2 cycle whereas the latter to a (limit) periodic or quasi-periodic cycle.

On the one hand, the flip bifurcation leads to a period-2 cycle that can be seen as the stationary solution of the initial dynamic system iterated at order 2.25 This

24A more structural approach will require to simulate and estimate the OLG model in the presence of

stochastic limit cycles (i.e., a deterministic limit cycle where the stochastic component is essentially an i.i.d. process). This could be done by determining the topological normal form for the flip (respectively, Hopf) bifurcation using Taylor expansions (see Kuznetsov [37]) or perturbation methods (e.g., Galizia ([28])). Note that the addition of shocks to our 2-sector OLG model will eliminate the perfect pre-dictability of the endogenous cyclical forces, but does not change the endogenous mechanisms explained before. We leave this issue for further research.

25For sake of illustration, and without loss of generality, consider the following (univariate) discrete

dynamical system:

x 7→ f (x; α) ≡ fα(x) (36)

where the map fα is invertible for small |α| in the neighborhood of the origin and the system has the

fixed point x0 = 0 for all α. Consider now the second iterate fα2(x) of the map (36). The map fα2 has

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iterated system usually leads to the resolution of a degree-2 polynomial that gives a high steady-state (above the initial stationary state) and a low steady-state (below the initial stationary state), i.e. the period-2 cycle.26 If we are in the period-2 cycle, every

two periods we return to one of the two steady-states of the second iterate of the initial dynamical system. In this respect, it turns out that the flip bifurcation, and thus the period-2 cycle, rests on the long-run component of the inheritance flows as a ratio of national income (or flows). Therefore it is critical to extract this component and to characterize its dynamics.

In the case of Hopf bifurcation the dynamics are different. Notably one obtains a periodic orbit that corresponds to a ”circle”. If one is on this circle, the dynamics are either periodic or quasi-periodic. It depends on the curvature coefficient (or stability index) that is obtained through a tedious derivation of the (topological) normal form of the Hopf bifurcation by means of smooth invertible coordinate and parameter changes. In particular, if this coefficient is rational then the dynamics on the circle are periodic, whereas if this coefficient is irrational, the dynamics on the circle are almost periodic, i.e. dense on the circle. From an empirical point of view, in the same spirit as Beaudry et al. [13], Charpe et al. [19], and Muck et al. [33], it means that one would expect a significant contribution of the medium-term component of the inheritance variable (after extracting the long-run component).27

xi = fα2(xi) i = 1, 2

that satisfy (period-2 cycle)

x2= fα(x1) and x1= fα(x2)

with x16= x2 and x1and x2are two stable or unstable equilibria depending on the sign of the coefficient

associated to the degree-3 term in a normal from expression of (36) (see footnote 26). For further technical details, see Chapters 4 and 9 of Kuznetsov ([37]).

26Under suitable regularity conditions, starting from the map (36), one can show that there exists a

topological normal form for the flip bifurcation defined by: η 7→ −(1 + β)η + sη3

where β = g(α) with g(0) = 0, and η is defined from successive changes of coordinates and parameters. On top of the trivial stationary solution η = 0, this leads to the resolution of a degree-2 polynomial. The equilibria x1 and x2are stable (unstable) if s = −1(+1). For further technical details, see Theorem 4.3.

of Kuznetsov ([37]).

27While there is no consensus, it is generally considered that the high-frequency component captures

the periodicity below 8 years, i.e. business fluctuations and noise, the medium-frequency component the periodicity between 8 and 30 to 50 years, and the low-frequency component the periodicity above 30 to 50 years.

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6.2

The long-run component of bequests

To assess some empirical relevance of period-2 cycles, we proceed in two steps. First, we consider the (stochastic) long-run component of the inheritance flows as a fraction of national income using the low-frequency approach initiated by M¨uller and Watson ([47], [48]) and a bandpass filter. Second, as in M¨uller and Watson ([47],[45]), we make use of some parametric models of persistence to characterize the dynamics of the long-run component.

In this respect, Figure 1 depicts the inheritance flow as a fraction of national income for France using annual data from 1897 through 2008, and three filter-based long-run components. The inheritance variables displays medium- to long-run swings: it was about 20–25% of national income between 1820 and 1910, down to less than 5% in 1950, and back up to about 15% by 2010.28 Notably, Figure 1 suggests a long-run cyclical behavior of inheritance flows.29. One key issue regards the stationarity of this series.

Standard stationary tests report mixed evidence, especially due to our sample size and the corresponding low power of unit root tests to distinguish between non-stationary and near-stationary stochastic processes.30 We further discuss this issue later on.

Using the structural breakpoint tests of Andrews ([1]) and Andrews and Ploberger ([2]) with a standard 15% sample trimming, both the exponential statistics and the simple average of the individual F-statistics provide no empirical support for the existence of a structural break over the period 1912-1991, when the data generating process of the observed series is assumed to be an autoregressive process of order one, AR(1).31.

Furthermore, the (sequential) multibreak points test of Bai and Perron ([5], [6], and [7]) report no evidence against the null hypothesis of no break when using a 15% trimming, a

28Using Eq. (1) of Piketty [51], the inheritance flow as a ratio of national income is defined by: Bt

Yt = µt× mt×

Wt

Yt

where Bt, Yt, Wt, µtand mt denote respectively the aggregate inheritance flow, the aggregate national

income, the aggregate private wealth, the mortality rate and the ratio between average wealth of the deceased and average wealth of the living. For further details, see Section 3 and the technical appendix of Piketty [51], , and Piketty and Zucman [53].

29A similar pattern is observed with UK data as shown by Atkinson [3]

30Appendix 8.12 reports the spectral density of the inheritance variable, with a typical hump shape and

a peak at low frequency. As a robustness analysis, we also transform our data using the first difference operator and apply the Baxter-King filter. This requires in turn to cumulate the filtered series. On the other hand, using the appropriate asymptotic results, the low-frequency approach of M¨uller and Watson ([47], [48]) remains valid for I(1) models. Therefore we consider the level specification as a more plausible way to extract the long-run component, possibly at the expense of a loss of efficiency for the Baxter-King filter.

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upper bound of five structural breaks, and a 5% significance level: the scaled F-statistics, which equals 5.53, is far below the critical value (11.47). In this respect, there is no compelling evidence that the inheritance ratio displays structural breaks or even some regime changes (e.g., a Markov switching model). However, the small sample size of our annual data might be an issue, as well as the absence of an incomplete long-run cycle in the observed variable.

Turning to the filtered series, the first long-run component is obtained from the ap-proximate low bandpass filter of Baxter and King ([10]).32 We filter the high- and

medium-frequency with periodicity below 40 years.33 On the other hand, the second trend is constructed using the methodology proposed by M¨uller and Watson ([47], [48]), namely by extracting long-run sample information after isolating a small number of low-frequency trigonometric weighted averages.34 In so doing, we project the series into a

constant and twelve (q) cosine functions with periods 2Tj for j = 1, · · · , 12 in order to capture the variability for periods longer than 20 years (2T /q). Importantly, one advan-tage of the low-frequency approach of M¨uller and Watson ([47], [48]) relative to bandpass or other moving average filters, is that it is applicable beyond the I(0) assumption.35

Finally, for sake of comparison, the third long-run component is obtained after adjusting a quadratic trend.

A quick eye inspection suggest that the bandpass and projection-based (using q = 6) trends contribute significantly to the total variability of the initial series, and display a nonlinear pattern. One main difference between these two measures is that the cosine transforms only use a limited number of points, which is consistent with the periodogram of the inheritance variable and the number of periodogram ordinates that fall into the low-frequency region. Intuitively, 112 years of data contain only limited information about long-run components with periodicities of more than 30 or 40 years.

To further highlight some statistical features of the inheritance variable, we make long-run inference regarding the mean, the standard deviation and conduct some persistence

32We also implement the asymmetric filter proposed by Christiano and Fitzgerald ([21], [20]) 33Qualitative conclusions remain robust when considering other periodicities.

34Let {x

t, t = 1, · · · , T } denote a (scalar) time series. Let Ψ(s) = [Ψ1(s), · · · , Ψq(s)]0 denote a Rq

-valued function with Ψj(s) =

2cos(jsπ), and let ΨT =Ψ 1−0.5T  , Ψ 2−0.5T  , · · · , Ψ T −0.5T

0 denote the T × q matrix after evaluating Ψ(.) at s = t−0.5T , for t = 1, · · · , T . The low-frequency projection is the fitted series from the OLS regression of [x1, · · · , xT] onto a constant and ΨT.

35For an extensive discussion about the relationship of this approach with spectral analysis, the scarcity

of low-frequency information, and the relevance of the approximation using a small q, see M¨uller and Watson ([47])

Figure

Figure 1: Inheritance and long-run components
Figure 2: Inheritance and short- to medium-run components
Figure 3: Spectral density of the inheritance variable
Figure 4: Spectral density

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